The product of two natural numbers is 18. If one is added to four times the smaller number this is equal to the larger number. Find both numbers
(4 * 2) +1 = 9
a = the smaller number
b = the larger number
The product of two natural numbers is 18.
This mean :
a * b = 18
If one is added to four times the smaller number this is equal to the larger number.
This mean :
1 + 4 a = b
So :
b = 1 + 4 a
Replace this value in expression :
a * b = 18
a * ( 1 + 4 a ) = 18
a + 4 a ^ 2 = 18
4 a ^ 2 + a = 18 Subtract 18 to both sides
4 a ^ 2 + a - 18 = 18 - 18
4 a ^ 2 + a - 18 = 0
The solutions are : a = - 9 / 4 and a = 2
- 9 / 4 isn't the natural number.
Solution: a = 2
b = 1 + 4 a = 1 + 4 * 2 = 1 + 8 = 9
The smaller number = 2
The larger number = 9
To find the two natural numbers, we can set up a system of equations based on the given information.
Let's assume the smaller number is x and the larger number is y.
We are given two pieces of information:
1) The product of the two numbers is 18:
x * y = 18
2) One is added to four times the smaller number is equal to the larger number:
1 + 4x = y
We can solve this system of equations by substitution or elimination:
Option 1: Substitution Method
First, solve equation 2) for y in terms of x:
y = 1 + 4x
Now, substitute this expression for y in equation 1):
x * (1 + 4x) = 18
Expand and simplify:
x + 4x^2 = 18
Rearrange the equation to a quadratic form:
4x^2 + x - 18 = 0
Now, we can factor or use the quadratic formula to solve for x. Factoring doesn't work easily in this case, so let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation:
a = 4, b = 1, c = -18
Substituting these values into the quadratic formula, we get:
x = (-1 ± √(1^2 - 4 * 4 * -18)) / (2 * 4)
x = (-1 ± √(1 + 288)) / 8
x = (-1 ± √289) / 8
x = (-1 ± 17) / 8
Since we are looking for natural numbers, we can discard the negative solution, and consider only the positive solution:
x = (17 - 1) / 8
x = 16 / 8
x = 2
Now that we have found the value of x, we can substitute it back into equation 2) to find y:
y = 1 + 4x
y = 1 + 4 * 2
y = 1 + 8
y = 9
Therefore, the two natural numbers that satisfy the given conditions are x = 2 and y = 9.